development of fuzzy process control charts and fuzzy unnatural pattern analyses

Computational Statistics & Data Analysis 51 (2006) 434–451www.elsevier.com/locate/csda

Development of fuzzy process control charts andfuzzy unnatural pattern analyses

Murat Gülbaya,b,∗, Cengiz Kahramana

aIndustrial Engineering Department, Istanbul Technical University, Istanbul, TurkeybIndustrial Engineering Department, University of Gaziantep, Gaziantep, Turkey

Available online 15 May 2006

Abstract

Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise obser-vation of a quantitative variable can be described by a special type of membership function defined on the set of all real numberscalled a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristicsare vague, uncertain, incomplete or linguistically defined. Fuzzy set theory is an inevitable tool for fuzzy control charts as well asother applications subjected to uncertainty in any form. The vagueness can be handled by transforming incomplete or nonprecisequantities to their representative scalar values such as fuzzy mode, fuzzy midrange, fuzzy median, or fuzzy average. Then crispmethods may be applied to those representative values for control chart decisions as “in control” or “out of control”. Transformingthe vague data by using one of the transformation methods may result in biased decisions since the information given by the vaguedata is lost by the transformation. Such data needs to be investigated as fuzzy sets without transformation, and the decisions basedon the vague data should not be concluded with an exact decision. A “direct fuzzy approach (DFA)” to fuzzy control charts forattributes under vague data is proposed without using any transformation method. Then, the unnatural patterns for the proposedfuzzy control charts are defined using the probabilities of fuzzy events.© 2006 Elsevier B.V. All rights reserved.

Keywords: Fuzzy process control; Run tests; Unnatural patterns; Fuzzy probability

1. Introduction

Control charts have been widely used for monitoring process stability and capability. Control charts are based ondata representing one or several quality-related characteristics of the product or service. If these characteristics aremeasurable on numerical scales, then variable control charts are used. If the quality-related characteristics cannot beeasily represented in numerical form, then attribute control charts are useful. When a process is in statistical control, acontrol chart displays the known patterns of variation. When the control chart points deviate from these known patterns,the process is considered to be out of control. The control chart distinguishes between normal and nonnormal variationthrough the use of statistical tests and control limits. The control limits are calculated using the rules of probability sothat when a point is determined to be out of control, it is due to an assignable cause and not due to a normal variation.

∗ Corresponding author. Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey. Tel.: +90 212 2931300x2073;fax: +90 212 2407260.

E-mail address: [email protected] (M. Gülbay).

0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2006.04.031

http://www.elsevier.com/locate/csda

mailto:[email protected]

M. Gülbay, C. Kahraman / Computational Statistics & Data Analysis 51 (2006) 434–451 435

A

A

B

C

C

B

CL

UCL

LCL

CL-1σ

CL-2σ

CL+2σ

CL+1σ

Fig. 1. The zones of a control chart.

The points outside the control limits are not the only criteria to determine the out of control conditions. However allpoints are inside the limits, the process may still be out of control if it does not display a normal pattern of variation.The zone tests, which are hypothesis tests in a modified form, are used to determine out of control conditions. They areused to test if the plotted points are following a normal pattern of variation. For a control chart to be effective, someaction must be taken as a result of the chart pattern. When the process average is centered where it is supposed to be,and the variability displays a normal pattern, the process is considered to be in control. A normal pattern means that theprocess is aligned with the probabilities of the normal distribution. A large abnormal variability and unnatural patternsindicate out of control conditions. Out of control conditions usually have assignable causes that must be investigated andresolved. Numerous supplementary rules, like zone tests or run rules have been developed to assist quality practitionersin detection of the unnatural patterns for the crisp control charts. The run rules are based on the premise that a specificrun of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this mustmean that something has changed in the process to produce a nonrandom or unnatural pattern.

The control charts may indicate an out-of-control condition when either one or more points fall beyond the controllimits or plotted points show some nonrandom patterns of behavior. Unnatural (nonrandom) patterns for classicalcontrol charts have been extensively studied. Over the years, many rules have been developed to detect nonrandompatterns within the control limits. Under the pattern-recognition approach, numerous researches have defined severaltypes of out-of-control patterns (e.g. trends, cyclic pattern, mixture, etc.) with a specific set of possible causes. Whena process exhibits any of these unnatural patterns, it implies that those patterns may provide valuable information forprocess improvement. The unnatural (nonrandom) patterns for fuzzy control charts have not been studied yet.

The zones of a control chart used in the zone tests are bounded by the standard deviations of the data as illustratedin Fig. 1. The probability of each zone based on the normal distribution is depicted in Fig. 2.

The main idea behind defining a rule for an unnatural pattern is the probability of the occurrence: these rules arebased on the premise that a specific run of data has a low probability of occurrence in a completely random stream ofdata. In general, probability of occurrence of an unnatural pattern is less than 1%. In the literature, there exist someunnatural patterns defined for the crisp cases. There is no certain rule about which unnatural patterns to use and theselection of a set of rules depends on the user preferences. Unnatural patterns are defined for the short runs, i.e., rulesfor a 15–20 consecutive points on the chart are investigated.

The Western Electric (1956) suggested a set of decision rules for detecting unnatural patterns on control charts.Specifically, it suggested concluding that the process is out of control if any of the following conditions is satisfied.

Rule 1: A single point falls outside of the control limits (beyond ±3� limits);Rule 2: Two out of three successive points fall in zone A or beyond (the odd point may be anywhere. Only two points

count).Rule 3: Four out of five successive points fall in zone B or beyond (the odd point may be anywhere. Only four points

count).Rule 4: Eight successive points fall in zone C or beyond.

436 M. Gülbay, C. Kahraman / Computational Statistics & Data Analysis 51 (2006) 434–451

A

2.14%

B

13.59 %

C

34.13 %

C

34.13 %

B

13.59 %

A

2.14 %

+3σ+2σ+1σ-1σ-2σ-3σ µ

Fig. 2. The zones and probabilities of normal distribution.

One-sided probabilities of the rules above are calculated as 0.00135, 0.0015, 0.0027, and 0.0039, respectively.Grant and Leavenworth (1988) recommended that nonrandom variations are likely to be presented if any one of the

following sequences of points occurs in the control charts.

Rule 1: Seven consecutive points on the same side of the center line.Rule 2: At least 10 of 11 consecutive points on the same side of the center line.Rule 3: At least 12 of 14 consecutive points on the same side of the center line.Rule 4: At least 14 of 17 consecutive points on the same side of the center line.One-sided probabilities of the rules above are calculated as 0.00781, 0.00586, 0.00647, and 0.00636, respectively.Nelson (1985) proposed the following rules for unnatural patterns:

Rule 1: One or more points outside of the control limits.Rule 2: Nine consecutive points in the same side of center line.Rule 3: Six points in a row steadily increasing or decreasing.Rule 4: Fourteen points in a row altering up and down.Rule 5: Two out of 3 points in a row in zone A or beyond.Rule 6: Four out of 5 points in zone B or beyond.Rule 7: Fifteen points in a row in zones C, above and below the centerline.Rule 8: Eight points in a row on both sides of the centerline with none in zone C.

The unnatural patterns tend to fluctuate too wide or they fail to balance around the centerline. The portrayal of naturaland unnatural patterns is what makes the control chart a very useful tool for statistical process and quality control.When a chart is interpreted, we look for special patterns such as cycles, trends, freaks, mixtures, groupings or bunchingof measurements, and sudden shifts in levels.

A product is generally classified in a binary manner (conforming or nonconforming) in attributed control charts.Binary classification may not be the most suitable if the product quality changes gradually rather than abruptly. Variousprocedures have been proposed for monitoring procedures in which the data are categorical in nature. In past research,Wang and Raz (1988, 1990), Raz and Wang (1990) and Kanagawa et al. (1993) proposed an assessment of intermediatequality level instead of the traditional binary judgement when the quality characteristics are not numerically measurable.Linguistic variables represent features by linguistic terms instead of numerical measurement. For example, the productquality feature can be classified by one of the terms ‘perfect’, ‘good’, ‘poor’, and ‘bad’, depending on the product’sdeviation from specifications. These words or phrases are called linguistic variables. Furthermore, each linguisticvariable value can be represented as a membership function. A linguistic variable differs from a numerical variable


in that its values are not numbers but words or phrases in a language (Wang and Raz, 1990). Linguistic variables arederived from human subjective judgements. Wang and Raz (1988, 1990) and Raz and Wang (1990) provided a potentialapplication of a fuzzy-set theory and constructed a control chart for these linguistic variables under the consideration ofa normal distribution. Kanagawa et al. (1993) provided another control chart under an estimated distribution function.Laviolette et al. (1995) pointed out that the control limit proposed by Wang and Raz (1990) had no satisfactory efficiencywhen it was used to detect a change in the process mean.

It is reasonable to treat the linguistic and uncertain data in the light of the fuzzy-set theory. Although the fuzzymethods provide a powerful framework for pattern recognition due to their ability to generate gradual membershipsof objects to clusters, a number of rules have been proposed to defuzzify the classification results in order to be ableto make a final (crisp) decision about the process. It is necessary to convert the fuzzy sets associated with linguisticvalues into scalars, which will be referred to as representative values. This conversion may be done in a number ofways, as long as the result is intuitively representative of the range of the base variable included in the fuzzy set, butthe concept of the information connected with the concept of uncertainty has been lost upon this conversion. Some ofthem are fuzzy mode, fuzzy median, and �-level fuzzy midrange (Zadeh, 1965). The purpose of data analysis is to gaininformation from data. By applying a conversion method to the vague data which result in scalar data we have alreadylost the information at the initial stage. Recently, �-level fuzzy control charts for attributes are proposed by Gülbay etal. (2004) in order to reflect the vagueness of the data and tightness of the inspection.

The objective of this paper is to develop a monitoring and diagnostic system to indicate natural out-of-controlsituations without defuzzification and define fuzzy unnatural patterns for the fuzzy control charts.

The paper is organized as follows: fuzzy process control charts based on defuzzification using fuzzy transformationmethods, and an alternative approach without any defuzzification are presented in Section 2. Probability of fuzzy eventsis explained in Section 3. Based on the probability of fuzzy events, fuzzy unnatural pattern rules for fuzzy control chartsare given in Section 4. A numerical illustrative example is presented in Section 5. Finally, concluding remarks are givenin Section 6.

2. Fuzzy process control charts

A fuzzy approach is suitable for attributes control charts (p, np, c, and u charts) when the data is linguistic, categorical,uncertain, or human dependent subjective judgement is possible.

In classical p charts, products are distinctly classified as “conformed” or “nonconformed” when determining fractionrejected. In fuzzy p control charts (See: Gülbay et al., 2004), when categorizing products, several linguistic termsare used to denote the degree of being nonconformed product such as “standard”, “second choice”, “third choice”,“chipped”, and so on… . A membership degree of being a nonconformed product is assigned to each linguistic term.Sample means for each sample group, Mj , are calculated as:

Mj =∑t

i=1 kij ri

mj

, (1)

where t is the total number of linguistic terms, kij the number of products categorized with the linguistic term i in thesample j , ri (0�ri �1) the membership degree of the linguistic term i, and mj the total number of products in samplej . Center line, CL, is the average of the means of the n sample groups and can be determined by

CL = Mj =∑n

j=1 Mj

n, (2)

where n is the number of sample groups initially available. kij and ri in Eq. (1), and so in Eq. (2), are the uncertainvalues and depend on the human subjective judgment. In another word, a sample can be belonged to the second choicecategory by a quality controller, while it may be included in the standard or third choice by another quality controller.In the same way, defining a membership degree for a category may depend on the quality controller preferences.Therefore, the value of Mj may lie between 0 and 1, as a result of these human judgments. It is clear that CL inEq. (2) has a range between 0 and 1 too. To overcome the uncertainty in the determination of the CL, fuzzy set theorycan successfully be adopted by defining CL as a triangular fuzzy number (TFN) whose fuzzy mode is CL, as shown in


�j (x)

CL = M = p

1

1 0

Lj(α)

Rj(α)

x

�

Fig. 3. TFN representation of M and Mj of the sample j .

Fig. 3. Then, for each sample mean, Lj (�) and Rj (�) can be calculated using Eqs. (3) and (4), respectively

Lj (�) = Mj�, (3)

Rj (�) = 1 − [(1 − Mj

)�]

. (4)

The membership function of the M , or CL, can be written as

�Mj(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 if x�0,

x

Mif 0�x�M,

1 − x

1 − Mif M �x�1,

0 if x�1.

(5)

The control limits for �-cut is also a fuzzy set and can be represented by TFNs. Since the membership function of CL isdivided into two components, then, each component will have its own CL, LCL, and UCL. The membership functionof the control limits depending upon the value of � is given in Eq. (6) and illustrated in Fig. 4

Control limits (�) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

CLL = M�

LCLL = max

⎧⎨⎩CLL − 3

√(CLL

) (1 − CLL

)n

, 0

⎫⎬⎭

UCLL = min

⎧⎨⎩CLL + 3

√(CLL

) (1 − CLL

)n

, 1

⎫⎬⎭

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

if 0�Mj �M,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

CLR = 1 − [(1 − M�

)�]

LCLR = max

⎧⎨⎩CLR − 3

√(CLR

) (1 − CLR

)n

, 0

⎫⎬⎭

UCLR = min

⎧⎨⎩CLR + 3

√(CLR

) (1 − CLR

)n

, 1

⎫⎬⎭

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

if M �Mj �1,

(6)

where n is the average sample size (ASS). When the ASS is used, the control limits do not change with the sample size.Hence, the control limits for all samples are the same.


LCLL (�)

LCLR (�)

UCLL (�)

UCLR(�)

CLR (�)

CLL (�)

M

α-level

1

Fig. 4. Illustration of the �-cut control limits.

1

a a� b c d� d

1

da a� d�b=cnumber of nonconformity

�

��

�

(a) (b)

Fig. 5. Representation of number of nonconformities by fuzzy numbers: (a) trapezoidal (a, b, c, d); (b) triangular (a, b, b, d).

For the variable sample size (VSS), n should be replaced by the size of the j th sample, nj . Hence, control limitschange for each sample depending upon the size of the sample. Therefore, each sample has its own control limits. Thedecision that whether process is in control (1) or out of control (0) for both ASS and VSS is as follows:

Process control ={

1 if LCLL(�)�Lj (�)�UCLL(�) ∧ LCLR(�)�Rj (�)�UCLR(�),

0 otherwise.(7)

The value of �-cut is decided with respect to the tightness of inspection such that for a tight inspection, � values closeto 1 may be used. As can be seen from Fig. 4, while � reduces to 0 (decreasing the tightness of inspection), the rangewhere the process is in control (difference between UCL and LCL) increases.

In the crisp case, the control limits for number of nonconformities are calculated by the Eqs. (8)–(10)

CL = c, (8)

LCL = c − 3√

c, (9)

UCL = c + 3√

c, (10)

where c is the mean of the nonconformities. In the fuzzy case, where number of nonconformity includes humansubjectivity or uncertainty, uncertain values such as “between 10 and 14” or “approximately 12” can be used todefine number of nonconformities in a sample. Then number of nonconformity in each sample, or subgroup, can berepresented by a trapezoidal fuzzy number (a, b, c, d) or a triangular fuzzy number (a, b, d) as shown in Fig. 5. Notethat a trapezoidal fuzzy number becomes triangular when b = c. For the ease of representation and calculation, atriangular fuzzy number is also represented as trapezoidal by (a, b, b, d) or (a, c, c, d).

Here, we propose a direct fuzzy approach (DFA) to deal with the vague data for the control charts. Transformingthe vague data by representing them with their representative values may result in biased decisions for particular dataespecially when they are represented by asymmetrical fuzzy numbers. The center line, CL, given in Eq. (8), is the mean


of the samples. For fuzzy case, where the numbers of nonconformities are represented by trapezoidal fuzzy numbers,fuzzy center line, C̃L, can be determined using the arithmetic mean of the fuzzy numbers and written as in Eq. (11)(see Chen and Hwang, 1992 for the fuzzy arithmetics performed in this paper)

C̃L =(∑n

j=1 aj

n,

∑nj=1 bj

n,

∑nj=1 cj

n,

∑nj=1 dj

n

)= (

a, b, c, d)

, (11)

where n is the number of fuzzy samples and a, b, c and d are the arithmetic means of the a, b, c, and d, respectively.C̃L can be rewritten as in Eq. (12). Then ˜LCL and ˜UCL are calculated using fuzzy arithmetics as given in Eqs. (13)and (14), respectively

C̃L = (a, b, c, d

)= (CL1, CL2, CL3, CL4) , (12)

˜LCL = C̃L − 3√

C̃L

=(CL1 − 3

√CL4, CL2 − 3

√CL3, CL3 − 3

√CL2, CL4 − 3

√CL1

)= (LCL1, LCL2, LCL3, LCL4) , (13)

˜UCL = C̃L + 3√

C̃L

=(CL1 + 3

√CL1, CL2 + 3

√CL2, CL3 + 3

√CL3, CL4 + 3

√CL4

)= (UCL1, UCL2, UCL3, UCL4) . (14)

An �-cut is a nonfuzzy set which comprises all elements whose membership is greater than or equal to �. Applying�-cuts of fuzzy sets (Fig. 5), values of a� and d� for samples and CL�

1 and CL�4 (start and end points of the �-cut of

CL) for center line are determined by Eqs. (15) and (16), respectively

a� = a + �(b − a), CL�1 = CL1 + � (CL2 − CL1) , (15)

d� = d − �(d − c), CL�4 = CL4 − � (CL4 − CL3) . (16)

Using �-cut representations, fuzzy control limits can be rewritten as given in Eqs. (17)–(19)

˜CL� = (CL�

1, CL2, CL3, CL�4

), (17)

˜LCL� = (LCL�

1, LCL2, LCL3, LCL�4

), (18)

˜UCL� = (UCL�

1, UCL2, UCL3, UCL�4

). (19)

The results of these equations can be illustrated as in Fig. 6. To retain the standard format of control charts and tofacilitate the plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with linguisticvalues into scalars referred to as representative values. This conversion may be performed in a number of ways as longas the result is intuitively representative of the range of the base variable included in the fuzzy set. The four ways, whichare similar in principle to the measures of central tendency used in descriptive statistics, are fuzzy mode, �-level fuzzymidrange, fuzzy median, and fuzzy average. It should be pointed out that there is no theoretical basis supporting anyone specifically and the selection between them should be mainly based on the ease of computation or preference ofthe user (Wang and Raz, 1990). The conversion of fuzzy sets into crisp values results in loss of information in linguisticdata. To retain the information of the linguistic data, we prefer to keep fuzzy sets as themselves and to compare fuzzysamples with the fuzzy control limits. For this reason, a direct fuzzy approach (DFA) based on the area measurementis proposed for the fuzzy control charts.

The decision about whether the process is in control can be made according to the percentage area of the samplewhich remains inside the ˜UCL and/or ˜LCL defined as fuzzy sets. When the fuzzy sample is completely involved bythe fuzzy control limits, the process is said to be “in-control”. If a fuzzy sample is totally excluded by the fuzzy controllimits, the process is said to be “out of control”. Otherwise, a sample is partially included by the fuzzy control limits. Inthis case, if the percentage area (�j ) which remains inside the fuzzy control limits is equal or greater than a predefinedacceptable percentage (�), then the process can be accepted as “rather in control”; otherwise it can be stated as “ratherout of control”. The possible decisions resulting from DFA are illustrated in Fig. 7. The parameters for determination


µ1

LCL1

LCL2

LCL3

LCL4

LCL1�

LCL4�

CL1�

CL4�

UCL4�

UCL�1

UCL4

UCL3

UCL2

CL4

UCL1

CL1

~

~

~

UCL

CL

LCL

CL2

CL3

α

Fig. 6. Representation of fuzzy control limits.

a b

c

d

~UCL

t1 t2 t1α

1

~LCL

Type U1 Type U2 Type U3 Type U4 Type U5 Type U6 Type U7

Type L1 Type L2 Type L3 Type L7 Type L6Type L4 Type L5

Fig. 7. Illustration of all possible sample areas outside the fuzzy control limits at �-level cut.


of the sample area outside the control limits for �-level fuzzy cut are LCL1, LCL2, UCL3, UCL4, a, b, c, d, and �.The shape of the control limits and fuzzy sample are formed by the lines of LCL1LCL2, UCL3UCL4, ab, and cd. Aflowchart to calculate area of the fuzzy sample outside the control limits is given in Fig. 8. The sample area above theupper control limits, AU

out, and the sample area falling below the lower control limits, ALout, are calculated. Equations

to compute AUout and AL

out are given in Appendix A. Then, the total sample area outside the fuzzy control limits, Aout,is the sum of the areas below the fuzzy lower control limit and above the fuzzy upper control limit. The percentagesample area within the control limits is calculated as given in Eq. (20)

��j = S�

j − A�out,j

S�j

, (20)

where S�j is the sample area at �-level cut.

The DFA provides the possibility of obtaining linguistic decisions like “rather in control” or “rather out of control”.Further intermediate levels of process control decisions are also possible by defining � in stages. For instance, it maybe defined as given below which is more distinguished

Process control =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

in control, 0.85��j �1,

rather in control, 0.60��j < 0.85,

rather out of control, 0.10��j < 0.60,

out of control, 0��j < 0.10.

(21)

The intermediate levels of process control decisions are subjectively defined by the quality expert. In binary classification(crisp case), the quality expert may only know if the process is in control or out of control. These predefined levelsmay refer to the truthful of the out of control. It can be used as a tracking and may give valuable information beforethe process is out of control. However intermediate levels are subjectively defined, it should refer to the depth ofinformation the quality expert needs to take some preventive actions.

3. Probability of fuzzy events

Analysis of fuzzy unnatural patterns for fuzzy control charts is necessary to develop. The formula for calculatingthe probability of a fuzzy event A is a generalization of the probability theory: in the case which a sample space X is acontinuum or discrete, the probability of a fuzzy event P(A) is given by (Yen and Langari, 1999):

P(A) ={∫

�A(x)PX(x) dx if X is continuous,∑i �A(xi)PX(xi) if X is discrete,

(22)

where PX denotes a classical probability distribution function of X for continuous sample space and probability functionfor discrete sample space, and �A is a membership function of the event A.

The membership degree of a fuzzy sample that belongs to a region is directly related to its percentage area falling inthat region, and therefore, it is continuous. For example, a fuzzy sample may be in zone B with a membership degreeof 0.4 and in zone C with a membership degree of 0.6. While counting fuzzy samples in zone B, that sample is countedas 0.4.

4. Generation of fuzzy rules for unnatural patterns

Numerous supplementary rules, like zone tests or run rules (Western Electric, 1956; Nelson, 1984, 1985; Duncan,1986; Grant and Leavenworth, 1988) have been developed to assist quality practitioners in detection of unnaturalpatterns for the crisp control charts. The run rules are based on the premise that a specific run of data has a lowprobability of occurrence in a completely random stream of data. If a run occurs, then this must mean that somethinghas changed in the process to produce a nonrandom or unnatural pattern. Based on the expected percentages in eachzone, sensitive run tests can be developed for analyzing the patterns of variation in the various zones.

For the fuzzy control charts, based on the Western Electric rules (1956), the following fuzzy unnatural pattern rulescan be defined. The probabilities of these fuzzy events are calculated using normal approach to binomial distribution.


Fig. 8. Flowchart to compute the area outside the fuzzy control limits.


The probability of each fuzzy rule (event) below depends on the definition of the membership function which issubjectively defined so that the probability of each of the fuzzy rules is as close as possible to the correspondingclassical rule for unnatural patterns. The idea behind this approach may justify the following rules.

Rule 1: Any fuzzy data falling outside the three-sigma control limits with a ratio of more than predefined percentage(�) of sample area at desired �-level. The membership function for this rule can subjectively be defined as below:

�1(x) =

⎧⎪⎨⎪⎩

0, 0.85�x�1,

(x − 0.60)/0.25, 0.60�x�0.85,

(x − 0.10)/0.50, 0.10�x�0.60,

1, 0�x�0.10.

(23)

Rule 2: A total membership degree around 2 from three consecutive points in zone A or beyond.Probability of a sample being in zone A (0.0214) or beyond (0.00135) is 0.02275. Let the membership function for

this rule be defined as follows:

�2(x) ={0, 0�x�0.59,

(x − 0.59)/1.41, 0.59�x�2,

1, 2�x�3.

(24)

Using the membership function above, the fuzzy probability given in Eq. (22) can be determined by Eq. (25)∫ 3

0�2(x)P2(x) dx =

∫ x1

0�2(x)P2(x) dx +

∫ x2

x1

�2(x)P2(x) dx +∫ 3

x2

�2(x)P2(x) dx

=∫ x2

x1

�2(x)P2(x) dx +∫ 3

x2

�2(x)P2(x) dx, (25)

where

PX(x) = PX

(z� x − np√

npq

). (26)

To integrate the equation above, the membership function is divided into sections each with a 0.05 width and �2(x)Px(x)

values for each section are added. For x1 = 0.59 and x2 = 2, the probability of the fuzzy event, rule 2, is determinedas 0.0015, which corresponds to the crisp case of this rule.

In the following rules, the membership functions are set in the same way.Rule 3: A total membership degree around 4 from five consecutive points in zone C or beyond:

�3(x) ={0, 0�x�2.42,

(x − 2.42)/1.58, 2.42�x�4,

1, 4�x�5.

(27)

The fuzzy probability for this rule is calculated as 0.0027.Rule 4: A total membership degree around 8 from eight consecutive points on the same side of the centerline with

the membership function below:

�4(x) ={

0, 0�x�2.54,

(x − 2.54)/5.46, 2.54�x�8.(28)

The fuzzy probability for the rule above is then determined as 0.0039.Based on Grant and Leavenworth’s rules (1988), the following fuzzy unnatural pattern rules can be defined.

Rule 1: A total membership degree around 7 from seven consecutive points on the same side of the center line. Thefuzzy probability of this rule is 0.0079 when membership function is defined as below:

�1(x) ={

0, 0�x�2.48,

(x − 2.48)/4.52, 2.48�x�7.(29)


Rule 2: At least a total membership degree around 10 from 11 consecutive points on the same side of the center line.The fuzzy probability of this rule is 0.0058 when the membership function is defined as below:

�2(x) ={0, 0�x�9.33,

(x − 9.33)/0.77, 9.33�x�10,

1, 10�x�11.

(30)

Rule 3: At least a total membership degree around 12 from 14 consecutive points on the same side of the center line.If the membership function is set as given below, then the fuzzy probability of the rule is equal to 0.0065

�3(x) ={0, 0�x�11.33,

(x − 11.33)/0.67, 11.33�x�12,

1, 12�x�14.

(31)

Rule 4: At least a total membership degree around 14 from 17 consecutive points on the same side of the center line.The probability of this fuzzy event with the membership function below is 0.0062

�4(x) ={0, 0�x�13.34,

(x − 13.34)/0.66, 13.34�x�14,

1, 14�x�17.

(32)

The fuzzy unnatural pattern rules based on Nelson’s Rules (1985) can be defined in the same way. Some of Nelson’srules (Rules 3 and 4) are different from the Western Electric Rules and Grant and Leavenworth’s rules. In order to applythese rules to fuzzy control charts, fuzzy samples can be defuzzified using �-level fuzzy midranges of the samples. The�-level fuzzy midrange, f �

mr, is defined as the midpoint of the ends of the �-cut. If a� and d� are the end points of �-cut,then,

f �mr = 1

2

(a� + d�) . (33)

Then Nelson’s 3rd and 4th rules are fuzzified as follows:Rule 3: Six points in a row steadily increasing or decreasing with respect to the desired �-level fuzzy midranges.Rule 4: Fourteen points in a row altering up and down with respect to the desired �-level fuzzy midranges.

5. An illustrative example

Samples of 200 units are taken every 4 h to control number of nonconformities. Data collected from 30 subgroupsshown in Table 1 are linguistic such as “approximately 30” or “between 25 and 30”. The linguistic expressions in Table 1are represented by fuzzy numbers as shown in Table 3. These numbers are subjectively identified by the quality controlexpert who also sets � = 0.60 and minimum acceptable ratio as � = 0.70. As � decreases, membership degree of thelinguistic terms decreases as well as the truthful of the decisions. Therefore, process control decisions may changeby choosing different levels of � and �. The quality control expert also set the acceptable membership degree ofunnaturalness as 0.95, that is, when a sample refers to an unnatural sample with respect to any rule, it should refer amembership degree of unnaturalness more than 0.95 with respect to the membership functions defined for that rule.

Using Eqs. (11)–(14), C̃L, ˜LCL, and ˜UCL are determined as follows:

C̃L = (18.13, 22.67, 26.93, 32.07),

˜LCL = (1.15, 7.10, 12.65, 19.29),

˜UCL = (30.91, 36.95, 42.50, 49.05).

Applying �-cut of 0.60, values of ˜CL�=0.60, ˜LCL�=0.60, and ˜UCL�=0.60 are calculated

˜CL�=0.60 = (20.85, 22.67, 26.93, 28.99),

˜LCL�=0.60 = (4.72, 7.10, 12.65, 15.31),

˜UCL�=0.60 = (34.53, 36.95, 42.50, 45.12).


Table 1Number of nonconformities for 30 subgroups

No. Approximately Between No. Approximately Between

1 30 16 402 20–30 17 32–503 5–12 18 394 6 19 15–215 38 20 286 20–24 21 32–357 4–8 22 10–258 36–44 23 309 11–15 24 25

10 10–13 25 31–4111 6 26 10–2512 32 27 5–1413 13 28 28–3514 50–52 29 20–2515 38–41 30 8

Table 2Fuzzy number (a, b, c, d) representation of 30 subgroups

No. a b c d f �=0.60mr No. a b c d f �=0.60

mr

1 25 30 30 35 28 16 33 40 40 44 37.22 15 20 30 35 18 17 28 32 50 60 30.43 4 5 12 15 4.6 18 33 39 39 43 36.64 3 6 6 8 4.8 19 12 15 21 38 13.85 32 38 38 45 35.6 20 23 28 28 36 266 16 20 24 28 18.4 21 28 32 35 42 30.47 3 4 8 12 3.6 22 14 18 28 33 16.48 27 36 44 50 32.4 23 24 30 30 34 27.69 9 11 15 20 10.2 24 20 25 25 31 23

10 7 10 13 15 8.8 25 25 31 41 46 28.611 3 6 6 10 4.8 26 7 10 25 28 8.812 27 32 32 37 30 27 3 5 14 20 4.213 11 13 13 15 12.2 28 23 28 35 38 2614 39 50 52 55 45.6 29 17 20 25 29 18.815 28 38 41 45 34 30 5 8 8 15 6.8

Average 18.13 22.67 26.93 32.07

Table 3Fuzzy zones calculated for the example

Zone a b c d

UCL� 34.53 36.95 42.50 45.12+2� 29.97 32.19 37.31 39.74+1� 25.41 27.43 32.12 34.37CL� 20.85 22.67 26.93 28.99−1� 15.47 17.48 22.17 24.43−2� 10.10 12.29 17.41 19.87LCL� 4.72 7.10 12.65 15.31

Based on the Western Electric Rules 1–4, membership functions in Eqs. (23), (24), (27), and (28) are used. Thesemembership functions set the degree of unnaturalness for each rule. As an example, when a total membership degreeof 1.90 is calculated for the rule 2, its degree of unnaturalness is determined from �2(x) as 0.9291.


Tabl

e4

Mem

bers

hip

degr

ees

offu

zzy

sam

ples

for

diff

eren

tzon

es(A

:Abo

ve,B

:Bel

ow)

No.

A+3

�in

+3�

B+3

�A

+2�

in+2

�B

+2�

A+1

�in

+1�

B+1

�A

CL

inC

LB

CL

A−1

�in

−1�

B−1

�A

−2�

in−2

�B

−2�

A−3

�in

−3�

B−3

�

10

01

00.

240.

760

10

0.94

0.06

01

00

10

01

00

20

01

00.

040.

960

0.38

0.62

0.25

0.52

0.23

0.64

0.36

00.

970.

030

10

03

00

10

01

00

10

01

00

10

0.18

0.82

00.

860.

144

00

10

01

00

10

01

00

10

01

00.

680.

325

01

00.

320.

680

10

01

00

10

01

00

10

06

00

10

01

00

10

0.54

0.46

0.27

0.73

00.

950.

050

10

07

00

10

01

00

10

01

00

10

01

00.

580.

428

0.13

0.73

0.14

0.61

0.39

00.

970.

030

10

01

00

10

01

00

90

01

00

10

01

00

10

0.05

0.95

00.

890.

110.

370.

630

100

01

00

10

01

00

10

01

00.

550.

450.

010.

990

110

01

00

10

01

00

10

01

00

10

0.74

0.26

120

01

00.

960.

040

10

10

01

00

10

01

00

130

01

00

10

01

00

10

01

01

00.

020.

980

141

00

10

01

00

10

01

00

10

01

00

150

0.98

0.02

0.56

0.44

01

00

10

01

00

10

01

00

160

10

0.72

0.28

01

00

10

01

00

10

01

00

170.

390.

390.

220.

650.

350

0.9

0.1

01

00

10

01

00

10

018

01

00.

490.

510

10

01

00

10

01

00

10

019

00

10

01

00.

030.

970

0.28

0.72

0.13

0.67

0.21

0.58

0.42

00.

970.

030

200

01

00.

050.

950

10

0.58

0.42

01

00

10

01

00

210

0.2

0.8

00.

990.

010.

610.

390

10

01

00

10

01

00

220

01

00

10

0.22

0.78

0.09

0.53

0.39

0.48

0.52

00.

870.

130

10

023

00

10

0.17

0.83

01

00.

890.

110

10

01

00

10

024

00

10

01

00.

20.

80

10

0.89

0.11

01

00

10

025

00.

510.

490.

280.

610.

10.

720.

280

10

01

00

10

01

00

260

01

00

10

0.01

0.99

00.

240.

760.

140.

420.

440.

430.

460.

110.

720.

280

270

01

00

10

01

00

10

0.01

0.99

00.

380.

620.

120.

760.

1228

00.

040.

960

0.53

0.47

0.27

0.73

00.

870.

130

10

01

00

10

029

00

10

01

00.

030.

970

0.63

0.37

0.39

0.61

00.

980.

020

10

030

00

10

01

00

10

01

00

10

0.02

0.98

01

0


Table 5Total membership degrees of the fuzzy samples in zones for the fuzzified Western Electric Rules

Sample no. Beyond ±3� In or above fuzzy CL In or below fuzzy CL

Rule 2 Rule 3 Rule 4 Rule 2 Rule 3 Rule 4

1 0.00 0.24 1 1 0 0 0.062 0.00 0.04 0.38 0.77 0.03 0.36 0.753 0.14 0 0 0 0.86 0.86 0.864 0.32 0 0 0 0.68 0.68 0.685 0.00 1 1 1 0 0 06 0.00 0 0 0.54 0.05 0.73 17 0.42 0 0 0 0.58 0.58 0.588 0.13 0.87 0.87 0.87 0 0 09 0.00 0 0 0 1 1 1

10 0.00 0 0 0 1 (µ = 1) 1 111 0.26 0 0 0 0.74 0.74 0.7412 0.00 0.96 1 1 0 0 013 0.00 0 0 0 1 1 114 1.00 0 0 0 0 0 015 0.00 1 1 1 0 0 016 0.00 1 (µ = 1)a 1 1 0 0 017 0.39 0.61 0.61 0.61 0 0 018 0.00 1 1 1 0 0 019 0.00 0 0.03 0.28 0.42 0.87 120 0.00 0.05 1 1 0 0 0.4221 0.00 0.99 1 1 0 0 022 0.00 0 0.22 0.61 0.13 0.52 0.9123 0.00 0.17 1 1 0 0 0.1124 0.00 0 0.2 1 0 0.11 125 0.00 0.9 1 1 0 0 026 0.00 0 0.01 0.24 0.57 0.86 127 0.12 0 0 0 0.88 0.88 0.8828 0.00 0.53 1 1 0 0 0.1329 0.00 0 0.03 0.63 0.02 0.61 130 0.00 0 0 0 1 1 1

aUnnatural sample with the corresponding degree of unnaturalness defined by the membership functions for each rule.

In order to make calculations easy and mine our sample database for unnaturalness a computer program is codedusing Fortran 90 programming language. Table 4 gives the total membership degrees of the fuzzy samples in variouszones.

The total membership degrees of the fuzzy samples (and degree of unnaturalness) for the fuzzified Western Elec-tric Rules are given in Table 5. Sample 14 shows an out of control situation with respect to Rule 1, while samples10 and 16 indicate an unnatural pattern with respect to the Rule 2. Their degrees of unnaturalness are determinedfrom the membership function of the Rule 2. As an example, considering Rule 2 of the fuzzified Western ElectricRules, the membership degrees of samples 15 and 16 become 2 and it corresponds to a degree of unnaturalness of 1(See Eq. (24)).

The total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules are repre-sented in Table 6.

As can be seen from Table 6, no samples indicate an unnatural pattern with respect to the fuzzified Grant andLeavenworth’s rules.

The rules 1, 2, 5, 6, 7, and 8 among Nelson’s rules can be examined in the same way. For Nelson’s Rules 3 and 4,the fuzzy samples are defuzzified by using �-level fuzzy midranges (given in Table 2) in order to check whether nextsample shows an increment or decrement or alternating. �-level fuzzy midranges for � = 0.60 are illustrated in Fig. 9,which refers no unnaturalness with respect to the Nelson’s Rules 3 and 4.


Table 6Total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules

Sample no. In or above CL In or below CL In or above CL In or below CL

Rule 1 Rule 2 Rule 3 Rule 4 Rule 1 Rule 2 Rule 3 Rule 4

1 1.00 0.06 – – – – – – – –2 0.77 0.75 – – – – – – – –3 0.00 0.86 – – – – – – – –4 0.00 0.68 – – – – – – – –5 1.00 0.00 – – – – – – – –6 0.54 1.00 – – – – – – – –7 0.00 0.58 3.31 – – – 3.93 – – –8 0.87 0.00 3.18 – – – 3.87 – – –9 0.00 1.00 2.41 – – – 4.12 – – –

10 0.00 1.00 2.41 – – – 4.26 – – –11 0.00 0.74 2.41 4.18 – – 4.32 6.67 – –12 1.00 0.00 2.41 4.18 – – 4.32 6.61 – –13 0.00 1.00 1.87 3.41 – – 4.32 6.86 – –14 0.00 0.00 1.87 3.41 5.18 – 3.74 6.00 7.67 –15 1.00 0.00 2.00 4.41 5.18 – 3.74 5.32 7.61 –16 1.00 0.00 3.00 4.41 5.41 – 2.74 5.32 6.86 –17 0.61 0.00 3.61 4.48 6.02 7.79 1.74 4.32 6.00 7.6718 1.00 0.00 4.61 5.48 7.02 7.79 1.00 3.74 5.32 7.6119 0.28 1.00 3.89 4.89 6.30 7.30 2.00 4.74 6.32 7.8620 1.00 0.42 4.89 5.89 6.76 8.30 1.42 4.16 5.74 7.4221 1.00 0.00 5.89 6.89 7.76 9.30 1.42 3.16 5.16 6.7422 0.61 0.91 5.51 7.51 7.51 8.92 2.34 3.34 6.08 7.6623 1.00 0.11 5.51 7.51 8.51 9.38 2.45 3.45 5.19 6.7724 1.00 1.00 5.90 8.51 9.51 10.38 3.45 3.45 5.19 7.1925 1.00 0.00 5.90 9.51 10.51 10.51 3.45 3.45 4.45 7.1926 0.24 1.00 5.86 8.75 9.75 10.75 3.45 4.45 5.45 7.1927 0.00 0.88 4.86 7.75 9.75 10.75 3.91 5.33 5.33 7.0728 1.00 0.13 4.86 8.14 10.75 11.75 4.04 5.46 5.46 6.4629 0.63 1.00 4.87 7.77 10.38 11.38 4.12 6.46 6.46 7.4630 0.00 1.00 3.87 7.49 9.38 11.38 5.01 6.46 7.46 7.46

Fig. 9. �-level (� = 0.60) fuzzy midranges of the fuzzy samples.


6. Conclusion

In the literature, numerous zone tests or run rules have been developed to assist quality practitioners in the detectionof unnatural patterns for the crisp control charts. The fuzzy control charts in the literature are commonly based on thefuzzy transformations to crisp cases. The unnatural patterns analyses for fuzzy control charts have not been studiedyet. In this paper, we have developed a direct fuzzy approach to fuzzy control charts without any defuzziffication,and then defined fuzzy unnatural pattern rules based on the probabilities of fuzzy events. The proposed fuzzy controlchart is illustrated with a numerical example. Then fuzzy unnatural pattern rules are developed and applied to the testproblem. We have defined fuzzy unnatural pattern rules based on the fuzzification of the crisp rules. The probability ofeach fuzzy unnatural pattern rule is calculated using the probability of fuzzy events. For further research, new fuzzyunnatural pattern rules can be developed and tested using fuzzy random variables. Useful and detailed informationabout the fuzzy random variables can be seen in Puri and Ralescu (1986).

Appendix A

Equations to compute sample area outside the control the limits

AUout = 1

2

[(d� − UCL�

4

)+ (dt − UCLt

4

)](max(t − �, 0))

+ 12

[(dz − az

)+ (c − b)](min(1 − t, 1 − �)), (A-U1)

where

t = UCL4 − a

(b − a) + (c − b)and z = max(t, �),

AUout = 1

2

[(d� − UCL�

4

)+ (c − UCL3)](1 − �), (A-U2)

AUout = 1

2

(d� − UCL�

4

)(max(t − �, 0)), (A-U3)

where

t = UCL4 − d

(UCL4 − UCL3) − (d − c),

AUout = 1

2

[(c − UCL3) + (

dz − UCLz4

)](min(1 − t, 1 − �)), (A-U4)

where

t = UCL4 − d

(UCL4 − UCL3) − (d − c)and z = max(t, �),

AUout = 1

2

[(dz2 − UCL

z24

)+ (dt1 − UCL

t14

)](min (max (t1 − �, 0) , t1 − t2))

+ 12

[(dz1 − az1

)+ (c − b)](min (1 − t1, 1 − �)) , (A-U5)

where

t1 = UCL4 − a

(b − a) + (UCL4 − UCL3), t2 = UCL4 − d

(UCL4 − UCL3) − (d − c),

z1 = max (�, t1) and z2 = max (�, t2)

AUout = 0, (A-U6)

AUout = 1

2

[(d� − a�)+ (c − b)

](1 − �), (A-U7)

ALout = 1

2

[(LCL�

1 − a�)+ (LCLt

1 − at)]

(max(t − �, 0))

+ 12

[(dz − az

)+ (c − b)](min(1 − t, 1 − �)), (A-L1)


where

t = d − LCL1

(LCL2 − LCL1) + (d − c)and z = max(�, t),

ALout = 1

2

[(d� − a�)+ (c − b)

](1 − �), (A-L2)

ALout = 1

2

[(LCL�

1 − a�)+ (LCL2 − b)](1 − �), (A-L3)

ALout = 1

2

[(LCL

z21 − az2

)+ (LCL

t11 − at1

)](min (max (t1 − �, 0) , t1 − t2))

+ 12

[(dz1 − az1

)+ (c − b)](min(1 − t, 1 − �)), (A-L4)

where

t1 = d − LCL1

(LCL2 − LCL1) + (d − c), t2 = a − LCL1

(LCL2 − LCL1) − (b − a),

z1 = max (�, t1) and z2 = max (�, t2) ,

ALout = 1

2

[(LCLz

1 − az)+ (LCL2 − b)

](min(1 − t, 1 − �)), (A-L5)

where

t = a − LCL1

(LCL2 − LCL1) − (b − a)and z = max(�, t),

ALout = 0, (A-L6)

ALout = 1

2

[(d� − a�)+ (c − b)

](1 − �). (A-L7)

References

Chen, S.-J., Hwang, C.-L., 1992. Fuzzy Multi Attribute Decision Making: Methods and Applications. Springer, Germany.Duncan, A.J., 1986. Quality Control and Industrial Statistics. fifth ed. Irwin Book Company, IL.Grant, E.L., Leavenworth, R.S., 1988. Statistical Quality Control. sixth ed. McGraw-Hill, New York.Gülbay, M., Kahraman, C., Ruan, D., 2004. �-cut fuzzy control charts for linguistic data. Int. J. Intell. Systems 19, 1173–1196.Kanagawa, A., Tamaki, F., Ohta, H., 1993. Control charts for process average and variability based on linguistic data. Int. J. Prod. Res. 31, 913–922.Laviolette, M., Seaman, J.W., Barrett, J.D., Woodall, W.H., 1995. A probabilistic and statistical view of fuzzy methods. Technometrics 37, 249–262.Nelson, L.S., 1984. The Shewhart control chart-tests for special causes. J. Qual. Technol. 16, 237–239.Nelson, L.S., 1985. Interpreting Shewhart x-bar control charts. J. Qual. Technol. 17, 114–116.Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422.Raz, T., Wang, J.H., 1990. Probabilistic and membership approaches in the construction of control charts for linguistic data. Prod. Plann. Control 1,

147–157.Wang, J.H., Raz, T., 1988. Applying fuzzy set theory in the development of quality control chart. Proceedings of the International Industrial

Engineering Conference, Orlando, FL, pp. 30–35.Wang, J.H., Raz, T., 1990. On the construction of control charts using linguistic variables. Int. J. Prod. Res. 28, 477–487.Western Electric, 1956. Statistical Quality Control Handbook. Western Electric, New York.Yen, J., Langari, R., 1999. Fuzzy Logic: Intelligence, Control, and Information. Prentice-Hall, NJ.Zadeh, L.A., 1965. Fuzzy set. Inform. Control 8, 338–353.

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